## On reference solutions and the sensitivity of the 2D Kelvin-Helmholtz instability problem.(English)Zbl 1442.76050

Summary: 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.

### MSC:

 76D17 Viscous vortex flows 76D05 Navier-Stokes equations for incompressible viscous fluids 76E20 Stability and instability of geophysical and astrophysical flows 76F20 Dynamical systems approach to turbulence 76F65 Direct numerical and large eddy simulation of turbulence 76M10 Finite element methods applied to problems in fluid mechanics

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