Rayleigh Taylor Instability Challenge

Introduction:

Rayleigh-Taylor instability (RTI) is the interpenetration of process that occurs whenever a light fluid pushes on a heavy fluid. It is a dynamic process whereby the two fluids seek to reduce their combined potential energy by rupturing plane between them. The ensuing turbulence and mixing has far-reaching consequences in many natural and man-made flows, ranging from supernovae to ocean waves.

Some practical CFD models that have been based on the R-T criteria are

1. Richtmyer-Meshkov Instability: occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instabilitybegins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and with spikes appearing in the case of a heavy fluid penetrating a light fluid. A chaotic regime eventually is reached and the two fluids mix. This instability can be considered the impulsive-acceleration limit of the Rayleigh–Taylor instability

2. Kelvin–Helmholtz instability (after Lord Kelvinand Hermann von Helmholtz) typically occurs when there is velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen when wind blows across a water surface; the instability constant manifests as waves. Kelvin-Helmholtz instabilities are also visible in the atmospheres of planets and moons, such as in cloud formations on Earth or the Red Spot on Jupiter, and the atmospheres (coronas) of stars, including the sun's.

3. Saffman–Taylor instability: also known as viscous fingering, is the formation of patterns in a morphologically unstable interface between two fluidsin a porous medium, described mathematically by Philip Saffman and  I. Taylor in a paper of 1958. This situation is most often encountered during drainage processes through media such as soils. It occurs when a less viscous fluid is injected, displacing a more viscous fluid; in the inverse situation, with the more viscous displacing the other, the interface is stable and no instability is seen. Essentially the same effect occurs driven by gravity (without injection) if the interface is horizontal and separates two fluids of different densities, the heavier one being above the other: this is known as the Rayleigh-Taylor instability. In the rectangular configuration the system evolves until a single finger (the Saffman–Taylor finger) forms, whilst in the radial configuration the pattern grows forming fingers by successive tip-splitting

Aim:

To simulate Rayleigh-Taylor instability, for water-air and water-user specified fluid.

 Geometry setup:

 Two rectangles were drawn of size 20x20mm in sketch command and they were 'pulled' to become surfaces. This is 2D simulation, hence surfaces were created. One of the rectangles would represent the containment of water and the other would be of air. As these are to be represented as different fluids, and mesh information needs to be shared in between, topology should be changed to 'share'.

That is all in geometry setup. Next, we move to mesh and physics setup.

Case 1: Base mesh size 0.5 mm

Animation:

https://drive.google.com/file/d/1oP8kNPdEO48jQfu7BZf4rdkjekg8k1df/view?usp=sharing

  Case 2: Base mesh size 1 mm

Animation :

https://drive.google.com/file/d/1GYEi12kmS7NAfro45JK7ojZCuCfY8qKU/view?usp=sharing

Case 3: Base mesh size 0.25 mm

Animation :

https://drive.google.com/file/d/1XFwK-YxVT_zLm1X49UmxIsLaRbAlggRJ/view?usp=sharing

Case 4:  simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh. 

Animation:

https://drive.google.com/file/d/10v56Ko3vXg76OQZjB7qnjQbYxTRLlrbH/view?usp=sharing

Conclusion :

Case 1,2 and 3 shows as the mesh size reduces, bubble formation seemed to be increased. Steady state simulation is used when the only end result of the simulation is important. Here, we are interested in the process of fluid interaction. Hence, transient simulation is used. Also, transient simulation gave freedom to use different time steps and Courant number as per the mesh size.

Atwood Number:

The Atwood number (A) is a dimensionless number used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as

 A = `(ρ1- ρ2)/(ρ1+ ρ2)`

where

ρ1 = density of the heavier fluid

ρ2= density of the lighter fluid

Atwood number is an important parameter in the study of Rayleigh–Taylor instability and Richtmyer–Meshkov instability. In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale, A.gt² where g is the gravitational acceleration and t is the time.