On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem
For researchers using Kelvin-Helmholtz instability as a benchmark, this work highlights fundamental limitations in achieving mesh-independent results due to inherent sensitivity.
This paper provides reference solutions for the 2D Kelvin-Helmholtz instability problem using high-order divergence-free finite element methods, but finds that mesh-independent prediction of final vortex pairing is not achievable due to sensitivity to small perturbations, explained via self-organization theory of 2D turbulence.
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.