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

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 meshindependent 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.

An explicit divergence-free DG method for incompressible flow

“…We observe the second merging process is completed at aroundt = 56, while the last merging process completed aroundt = 160, and at timet = 200 a single vortex is left. Comparing with the reference data [28], computed using an IMEX SBDF2, P 8 divergence-conforming HDG scheme [28] on a 256 × 256 uniform square mesh with time step size ∆t = δ 0 ×10 −3 ≈ 3.6×10 −5 , we observe quite a good agreement of the vorticity dynamics up to timet = 56 where the second merging process is completed. However, the numerical results in [28] show that the last merging appears in a much later time, at aroundt = 250.…”

“…Comparing with the reference data [28], computed using an IMEX SBDF2, P 8 divergence-conforming HDG scheme [28] on a 256 × 256 uniform square mesh with time step size ∆t = δ 0 ×10 −3 ≈ 3.6×10 −5 , we observe quite a good agreement of the vorticity dynamics up to timet = 56 where the second merging process is completed. However, the numerical results in [28] show that the last merging appears in a much later time, at aroundt = 250. The numerical dissipation in our simulation triggered the last vortex merging in a much earlier time, since we use a lower order method on a coarser mesh compared with [28].…”

“…In Fig. 5, we plot the evolution of kinetic energy and enstrophy of our simulation, together with the reference data provided in [28]. A good agreement of the kinetic energy can be clearly seen, while the enstrophy agrees pretty well till timet = 150, where the last vortex merging toke place for our simulation, while that happens at a much later timet = 250 for the scheme used in [28].…”

“…Example 3: Kelvin-Helmholtz instability problem. We consider the Kelvin-Helmholtz instability problem, the set-up is taken from [28]. The Navier-Stokes equations (16) with Reynolds number Re = 10000 on the domain [0, 1] × [0, 1] with a periodic boundary condition on the x-direction, and the free-slip boundary condition u · n = 0, ν((∇u)n) × n = 0 at y = 0 and y = 1.…”

“…The numerical dissipation in our simulation triggered the last vortex merging in a much earlier time, since we use a lower order method on a coarser mesh compared with [28]. We notice that a numerical simulation at the scale of [28] is out of reach for our desktop-based simulation. In Fig.…”