Taylor-Couette Instability in General Manifolds: A Lattice Kinetic Approach

TL;DR

This paper introduces a lattice kinetic method for simulating fluid flow in complex curved geometries, validated through critical Reynolds number calculations for various shapes, aligning well with theoretical and experimental data.

Contribution

The paper develops a novel lattice kinetic approach for fluid dynamics in general manifolds using contravariant coordinates and Hermite polynomial expansions, extending simulation capabilities.

Findings

Accurately predicts critical Reynolds numbers for Taylor-Couette instability in cylinders and spheres.

Finds a 10% increase in critical Reynolds number for tori compared to cylinders.

Demonstrates good agreement with existing theory and experimental data.

Abstract

We present a new lattice kinetic method to simulate fluid dynamics in curvilinear geometries. A suitable discrete Boltzmann equation is solved in contravariant coordinates, and the equilibrium distribution function is obtained by a Hermite polynomials expansion of the Maxwell-Boltzmann distribution, expressed in terms of the contravariant coordinates and the metric tensor. To validate the model, we calculate the critical Reynolds number for the onset of the Taylor-Couette instability between two concentric cylinders, obtaining excellent agreement with the theory. In order to extend this study to more general geometries, we also calculate the critical Reynolds number for the case of two concentric spheres, finding good agreement with experimental data. In the case of two concentric tori, we have found that the critical Reynolds is about 10% larger than the respective value for the two concentric cylinders.