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A Finite Volume Scheme for Savage-Hutter Equations on Unstructured Grids

Ruo Li, Xiaohua Zhang

TL;DR

A Godunov-type finite volume scheme on unstructured triangular grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate and the direct observation that the model is a not Galilean invariant system is shown.

Abstract

A Godunov-type finite volume scheme on unstructured triangular grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate. We show the direct observation that the model is a not Galilean invariant system. At the cell boundary, the modified Harten-Lax-van Leer (HLL) approximate Riemann solver is adopted to calculate the numerical flux. The modified HLL flux is not troubled by the lack of Galilean invariance of the model and it is helpful to handle discontinuities at free interface. Rigidly the system is not always a hyperbolic system due to the dependence of flux on the velocity gradient. Even though, our numerical results still show quite good agreements to reference solutions. The simulations for granular avalanche flows with shock waves indicate that the scheme is applicable.

A Finite Volume Scheme for Savage-Hutter Equations on Unstructured Grids

TL;DR

A Godunov-type finite volume scheme on unstructured triangular grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate and the direct observation that the model is a not Galilean invariant system is shown.

Abstract

A Godunov-type finite volume scheme on unstructured triangular grids is proposed to numerically solve the Savage-Hutter equations in curvilinear coordinate. We show the direct observation that the model is a not Galilean invariant system. At the cell boundary, the modified Harten-Lax-van Leer (HLL) approximate Riemann solver is adopted to calculate the numerical flux. The modified HLL flux is not troubled by the lack of Galilean invariance of the model and it is helpful to handle discontinuities at free interface. Rigidly the system is not always a hyperbolic system due to the dependence of flux on the velocity gradient. Even though, our numerical results still show quite good agreements to reference solutions. The simulations for granular avalanche flows with shock waves indicate that the scheme is applicable.

Paper Structure

This paper contains 12 sections, 1 theorem, 40 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Numerical Method
  4. Finite volume method
  5. HLL approximate Riemann solver
  6. Linear reconstruction
  7. Time stepping
  8. Numerical Examples
  9. Dam break test cases with exact solution
  10. Avalanche flows slide down an inclined plane and merge continuously into a horizontal plane
  11. Granular flow pasts obstacle
  12. Conclusions

Key Result

Theorem 1

For any unit vector $\bm{n}=(\cos{\theta}, \sin{\theta})$, $\theta \neq 0$, and all vectors $\bm{U}$, the following equality holds if and only if $\beta_x = \beta_y$, where the rotation matrix $\bm{T}$ is as

Figures (7)

  • Figure 1: The sketch of curvilinear coordinate for Savage-Hutter model(reproduced from Chiou2005).
  • Figure 2: A schematic of the ENO-type reconstruction on patch of cells (reproduced from Deng2013).
  • Figure 3: Exact and numerical solutions for the 1D dam break problem at $t=0.0,0.1,0.2,0.3,0.4$ and $0.5$
  • Figure 4: Thickness contours of the avalanche at eight different dimensionless times $t=3, 6, 9, 12, 15, 18, 21, 24$ for the flow slides down the inclined plane and merging continuously into a horizontal plane. The transition zone from the inclined plane to the horizontal plane lies between the two green dashed lines.
  • Figure 5: The triangular meshes at times $t=6, 12, 18, 24$ for the flow slides down the inclined plane and merging continuously into a horizontal plane.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof