Numerical computation of triangular cavity flows by a Lagrange-Galerkin scheme with a locally linearized velocity

TL;DR

This paper presents a stable and quadrature-free Lagrange-Galerkin scheme for simulating triangular cavity flows, demonstrating its effectiveness through numerical results including bifurcation and streamline patterns.

Contribution

We develop a locally linearized velocity Lagrange-Galerkin scheme that avoids numerical quadrature, ensuring stability and convergence in triangular cavity flow simulations.

Findings

Stable numerical scheme for cavity flows

Observation of bifurcation in stationary solutions

Streamline pattern analysis

Abstract

We show numerical results of triangular cavity flow problems solved by a Lagrange-Galerkin scheme free from numerical quadrature. The scheme has recently developed by us, where a locally linearized velocity and the backward Euler approximation are used in finding the position of fluid particle at the previous time step. Since the scheme can be implemented exactly as it is, the theoretical stability and convergence results are assured, while the conventional Lagrange-Galerkin schemes may encounter the instability caused by numerical quadrature errors. The scheme is employed to solve cavity flow problems in triangular domains, where we observe the bifurcation of stationary solutions and the patterns of streamlines.