Snapshot: Simulation of a Kelvin-Helmholtz instability using second order Finite Volume schemes and fourth order Discontinuous Galerkin methods

We present in-viscid and viscous simulations of a Kelvin-Helmholtz instability using second a order accurate monotoniced-central finite volume (FV) method and a fourth order accurate discontinuous Galerkin (DG) method. The initial condition is given by [1]:

$$\rho (t=0) = \frac{1}{2}
+ \frac{3}{4} B,
~~~~~~~~~
p (t=0) = 1,~~~~~~~~~~
$$

$$
v_1 (t=0) = \frac{1}{2} \left( B-1 \right),
~~~~~~~
v_2 (t=0) = \frac{1}{10} \sin(2 \pi x),
$$

with $$B=\tanh \left( 15 y + 7.5 \right) – \tanh(15y-7.5).$$

We first present the FV results at end time $t=3.7$, which are computed using a monotoniced-central second order discretization of the Euler equations of gas dynamics on uniform grids.

The next results use a fourth order DG discretization of the Navier-Stokes equations on uniform grids using $Re=320.000$ at end time $t=3.7$. The highest resolution (4096² DOFs) is a direct numerical simulation (DNS) of the problem, where all scales are resolved.

It is remarkable that the numerical dissipation of the second order FV scheme causes the in-viscid simulation with 2048² DOFs to look very similar to the viscous DNS solution at $Re=320.000$.

[1] A.M. Rueda-Ramírez, G.J Gassner (2021). A Subcell Finite Volume Positivity-Preserving Limiter for DGSEM Discretizations of the Euler Equations. https://arxiv.org/pdf/2102.06017.pdf