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Semi-Implicit Lagrangian Voronoi Approximation for Compressible Viscous Fluid Flows

Ondřej Kincl, Ilya Peshkov, Walter Boscheri

Abstract

This paper contributes to the recent investigations of Lagrangian methods based on Voronoi meshes. The aim is to design a new conservative numerical scheme that can simulate complex flows and multi-phase problems with more accuracy than SPH (Smoothed Particle Hydrodynamics) methods but, unlike diffuse interface models on fixed grid topology, does not suffer from the deteriorating quality of the computational grid. The numerical solution is stored at particles, which move with the fluid velocity and also play the role of the generators of the computational mesh, that is efficiently re-constructed at each time step. The main novelty stems from combining a Lagrangian Voronoi scheme with a semi-implicit integrator for compressible flows. This allows to model low-Mach number flows without the extremely stringent stability constraint on the time step and with the correct scaling of numerical viscosity. The implicit linear system for the unknown pressure is obtained by splitting the reversible from the irreversible (viscous) part of the dynamics, and then using entropy conservation of the reversible sub-system to derive an auxiliary elliptic equation. The method, called SILVA (Semi-Implicit Lagrangian Voronoi Approximation), is validated in a variety of test cases that feature diverse Mach numbers, shocks and multi-phase flows.

Semi-Implicit Lagrangian Voronoi Approximation for Compressible Viscous Fluid Flows

Abstract

This paper contributes to the recent investigations of Lagrangian methods based on Voronoi meshes. The aim is to design a new conservative numerical scheme that can simulate complex flows and multi-phase problems with more accuracy than SPH (Smoothed Particle Hydrodynamics) methods but, unlike diffuse interface models on fixed grid topology, does not suffer from the deteriorating quality of the computational grid. The numerical solution is stored at particles, which move with the fluid velocity and also play the role of the generators of the computational mesh, that is efficiently re-constructed at each time step. The main novelty stems from combining a Lagrangian Voronoi scheme with a semi-implicit integrator for compressible flows. This allows to model low-Mach number flows without the extremely stringent stability constraint on the time step and with the correct scaling of numerical viscosity. The implicit linear system for the unknown pressure is obtained by splitting the reversible from the irreversible (viscous) part of the dynamics, and then using entropy conservation of the reversible sub-system to derive an auxiliary elliptic equation. The method, called SILVA (Semi-Implicit Lagrangian Voronoi Approximation), is validated in a variety of test cases that feature diverse Mach numbers, shocks and multi-phase flows.

Paper Structure

This paper contains 22 sections, 2 theorems, 103 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Voronoi tessellation
  3. Basic properties and generation
  4. Discrete derivative operators on Voronoi tessellations
  5. Compressible Navier-Stokes system on a moving Voronoi mesh
  6. Time marching scheme
  7. Semi-implicit reversible step
  8. Explicit irreversible step
  9. Mesh relaxation step
  10. Multi-phase projection
  11. Boundary conditions
  12. Numerical results
  13. Gresho vortex
  14. Sedov problem
  15. Saltzman piston problem
  16. ...and 7 more sections

Key Result

Theorem 1

Let the computational domain $\Omega \in \mathbb{R}^3$ be assigned with periodic boundaries. Let $\bm{x}_i$, $\rho_i$, $\bm{v}_i$, $e_i$ obey eq:path-eq:boe. Define (These variables represent the total mass, momentum and energy of each cell respectively.) Then, the masses $M_i$ are constant and moreover, it holds true that In other words, the total momentum (both linear and angular) and energy a

Figures (15)

  • Figure 1: Two Voronoi cells $\omega_i$ and $\omega_j$ with our notation for edges, seed generators and edge midpoints.
  • Figure 2: An example of low-quality Voronoi mesh. Some cells are strongly elongated and some pairs of generating seeds almost overlap.
  • Figure 3: The velocity color map of the Gresho vortex at $t=0$ and $t=1$. A group of quasi-Lagrangian cells is highlighted in green. The Voronoi grid consists of 200x200 cells.
  • Figure 4: The $y$ component of the velocity along the positive $x$-axis at $t=3$. Results for various Mach numbers are shown. The error peaks at about $2\%$, which is an excellent agreement. For $\mathrm{Ma} = 0.001$, the stiffened gas equation of state was used.
  • Figure 5: The color plot of density field in the Sedov test.
  • ...and 10 more figures

Theorems & Definitions (5)

  • Theorem 1: Discrete conservation laws
  • proof
  • Theorem 2: Discrete entropic inequality
  • proof
  • Remark 1