# Kelvin Helmholtz instability (3D)

A 3D Kelvin-Helmholtz instability, describing the blending of two immiscible fluids with a shearing motion, is simulated at high resolution. The aim is to compute the pattern of the interface between the fluids. This is done first through a continuum advection equation, and then by injecting discrete particles. Due to the numerical diffusion in the continuum equation, much sharper figures are obtained with the particles.

The computational domain is the unit cube. Periodic boundary conditions are used all over the boundary of the computational domain. The Reynolds number is 7'500 with respect to the side length of the simulation. The equations solved are the full 3D incompressible Navier-Stokes equations for the motion of the fluid Initially, the computational domain is divided in three parts. The initial velocity of the middle part is (0.5, 0, 0) and of the other two parts is (-0.5, 0, 0) in a Cartesian reference frame. Random small-amplitude perturbations are added to the x and y components of the initial velocity field at the middle part of the domain, to trigger the development of the Kelvin-Helmholtz instability. With this setup, the coupled system is left to evolve. The grid used consists of 134'217'728 lattice nodes.

Two approaches are chosen to visualize the flow: a convection-diffusion equation with low diffusion (left side of the animation), or a large number of Lagrangian particles (right side of the animation). In the convection-diffusion equation, the Prandtl number is equal to 1.0

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It is seen that the discrete particles produce a much sharper picture than the convection-diffusion equation, because even when a low parameter of diffusion is chosen, this continuum mode is plagued by numerical diffusivity. The quality of the blending pattern with Lagrangian particles is even further illustrated in the animation below, which shows a 2D cut through the simulation:

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