# On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem

## TL;DR

This paper investigates the sensitivity of 2D Kelvin-Helmholtz instability simulations to small perturbations, providing high-order reference solutions and explaining the variability through turbulence self-organization theory.

## Contribution

It offers high-order divergence-free finite element reference solutions and a theoretical explanation for the problem's sensitivity to perturbations.

## Key findings

- Reference solutions obtained for multiple Reynolds numbers.
- Final vortex pairing prediction is mesh-independent only under certain conditions.
- Sensitivity to small perturbations explained by turbulence self-organization theory.

## Abstract

Two-dimensional Kelvin-Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin-Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed.