A Joint Eulerian-Lagrangian Method for the Solution of Multi-Scale Flow Problems

Abstract

Numerical convection schemes in a time-dependent Computational Fluid Dynamics simulation suffer from numerical diffusion, where a transported scalar quantity experiences a total diffusivity greater than any physical diffusivity due to the viscosity of the surrounding fluid. This work aims to develop and test a novel convection method, combining the Eulerian and Lagrangian frameworks, to eliminate numerical diffusion.

A conserved scalar quantity is decomposed into low- and high-frequency components. The low-frequency field is transported in the Eulerian framework using a high-order Central Differencing Scheme, which has a negligible numerical diffusivity and good cost efficiency, but exhibits oscillatory behaviour around sharp changes in gradient. The high-frequency information is described with (computationally expensive) massless Lagrangian particles, where the prescribed particle 'density' provides a balance between accuracy and computational cost. Particles are convected by interpolating the underlying velocity field on to the particle position, while particle diffusion is described using a stochastic Wiener process. After transport an Eulerian representation of the Lagrangian particle field is constructed and added to the low-frequency scalar component, to recover the overall transported field.

Formulations of the joint Eulerian-Lagrangian method for Direct Numerical Simulation and Large-Eddy Simulation are proposed. Re-initialisation (particle addition) and particle removal are implemented to maintain accuracy and to reduce the computational cost of the method. The accuracy of the Lagrangian reconstruction is improved through the development and application of localised filtering and deconvolution algorithms. The method is applied in two and three dimensions, where it is effective in removing numerical diffusion, but introduces noise into the scalar field due to the point-like nature of the particles. While the method is at least twice as expensive as traditional Eulerian simulations at the same grid resolution, it is capable of delivering better accuracy and considerably greater cost efficiency than Eulerian simulations at higher resolutions.