Rayleigh Taylor Instability Challenge

INTRODUCTION

The Rayleigh-Taylor instability or RT instability is the instability of an interface between two fluids when the heavier fluid is on top of the lighter fluid. The RT waves are found in many phenomena where a denser fluid is accelerated into a rarer fluid. The main examples are the mushroom cloud formed during a volcanic explosion or a nuclear blast.

AIM

The aim of this study is to perform a simulation of RT instability for different mesh sizes for air and water and also do study the behaviour of RT waves in Liquid Ammonia and water.

It is also important to define the Atwood number in the two cases.

CFD Models based on RT waves

Rayleigh-Taylor instability or the RT instability occurs in many natural as well as manmade phenomena such as the inertial confinement fusion, cavitation, combustion as well as astrophysical and geophysical flows. DNS and LES are used for developing significant knowledge about RT instability but the high computational cost for simulating high Reynolds number flows forced for the development for modelling RT instability.

The computational models used are :

K-L model

It is a two-equation model which is used to describe the turbulent regime of an RT and RM induced mixing. The equations are derived from buoyancy-drag models for the self-similar growth of RT and RM instabilities.

K-L-a model

It is a three equation model. It is an extension of the K-L model by including an extra equation for turbulent mass flux velocity.

K-L-a-b model

It is the four equation model which is an extension of the K-L-a model with the addition of the evolution equation for the density-specific volume covariance.

BHR-2 model

SIMULATION

The onset of the RT instability is analyzed for different mesh sizes and the effect of the grid size on the pattern formation is studied. For the simulation of multiple phases volume of fluid method is used. An implicit formulation is used to obtain a stable solution.

The geometry used for the Analysis is given below:

The simulation is run for different mesh sizes. The meshes are squares of sides 0.5 mm, 0.3 mm and 0.2 mm. The different meshes and the animations created by running the simulations are shown below.

The mesh of size 0.5 mm is shown below:

The residual plot is shown below:

The animation is shown below:

For the mesh size of 0.3 mm the mesh is shown below:

The residual plot is shown below for the 0.3 mm mesh:

The animation for 0.3 mm grid size is shown below:

The mesh of 0.2 mm grid size is shown below:

The animation for 0.2 mm grid size is shown below:

From the simulations, it can be seen that the patterns formed changes as the size of the grid is reduced. A grid independent study must be carried out to find out the perfect grid size for the simulation. As the grid size is reduced the time step also needs to be reduced which increased the running time.

It is important to note that a transient simulation is used instead of a steady-state simulation since we are interested in the time-dependent change that happens in the RT instability rather than the final state.

Atwood's Number (A):

It is a dimensionless number used in the study of instabilities at the interface for fluids with different densities.

It is a dimensionless density ratio defined as `A = \frac{rho_1 - rho_2}{rho_1 + rho_2}`

where,

`rho_1`=Density of water = `997 \frac{kg}{m^3}`

`rho_2`=Density of air = `1.225 \frac{kg}{m^3}`

A = (997 - 1.225 )/(997 + 1.225 ) = 0.9975 .

CONCLUSION

From the simulations, it can be seen that the pattern formed depends on the size of the grid. For exactly capturing the patterns formed we must use a very fine grid.

Atwood number is used for studying the instabilities in the interface of different fluids.