Rayleigh-Taylor instability

Rayleigh-Taylor Instability model

1. Rayleigh-Taylor instability (RTI) is the interpenetration of materials that occurs whenever the heavier liquid pushes onto the lighter liquid due to the presence of gravity. It is a dynamic process whereby the two fluids seek to reduce their combined potential energy. The ensuing turbulence and mixing has far-reaching consequences in many natural and man-made flows, ranging from supernovae to inertial confinement fusion capsules.

As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing \"plumes\" flowing upwards (in the gravitational buoyancy sense) and \"spikes\" falling downwards. In the linear phase, the fluid movement can be closely approximated by linear equation, and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large a linear approximation, and non-linear equations are required to describe fluid motions. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). The difference in the fluid densities divided by their sum is defined as the Atwood number, A. For A close to 0, RT instability flows take the form of symmetric \"fingers\" of fluid; for A close to 1, the much lighter fluid \"below\" the heavier fluid takes the form of larger bubble-like plumes.

practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves.

Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins 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.

Kelvin–Helmholtz instability can occur when there is velocity in a single continous fluid, or where there is a velocity difference across the interface between two fluids. An example is wind blowing over water: The instability manifests in waves on the water surface. More generally, clouds, the ocean, Saturn\'s bands and the sun\'s corona show this instability.

The theory predicts the onset of instability and transition to turbulent in fluids of different densities moving at various speeds. Helmholtz studied the dynamics of two fluids of different densities when a small disturbance, such as a wave, was introduced at the boundary connecting the fluids.

For some short enough wavelengths, if surface tension is ignored, two fluids in parallel motion with different velocities and densities yield an interface that is unstable for all speeds. Surface tension stabilises the short wavelength instability however, and theory predicts stability until a velocity threshold is reached. The theory, with surface tension included, broadly predicts the onset of wave formation in the important case of wind over water.

Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh Taylor Instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of ink jet technology. whereby a jet of liquid is perturbed into a steady stream of droplets.

The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface-tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions.

2. Performing the Rayleigh Taylor instability simulation for 3 different mesh sizes(0.5, 0.3, 0.2mm).

Geometry:

It is similar for all the cases. There are two boxes present one above other. Dimensions of each box is 20*20mm. Eventhough the geometry is similar we will be using different mesh operations each case.

Meshing:

Case-1(0.5mm): In this case, base mesh is opted which has a size of 0.5mm. This mesh size creates 2400 cells in the domain.

Case-2(0.3mm): In this case, mesh is refined to size of 0.3mm. This mesh size creates 4000 cells in the domain.

Case-3(0.2mm): In this case, mesh is refined further to size of 0.2mm. This mesh size creates 6000 cells in the domain.

Case setup:

Setup is same for all the cases.

Solver - transient, pressure based , absolute.

Steady state solver which define the behavior of the system or the process that are unchanging in time. In this case of this challenge behaviour of the fluids in the domain is completely unpredictable and so we used transient state which tells us when a process variable or variables have been changed and the system has not yet reached a steady state.The time taken for the solution to change from one steady state to another steady state is called the transient time.

Gravity is activated and the value -9.81m/s is given in the y-direction.

Models- Multi-zone, implicit solver.

Materials - water and air.

Phases- Primary- air

secondary - water.

Standard initialization.

Once the initialization is done then we have to add patches in the initialization region.

Initialization-patch-water-volume fraction-water domain-value\'1\'-patch.

patch-water-volume fraction-air domain-value\'0\'-patch.

Run parameters - timestep size-0.005 sec, no of timesteps-200.

Now the case setup is complete.

Solution results:

case-1: Below plot represents the residuals. Convergence is checked so the solution has stopped way before it has reached the 200 timesteps.

It is the contour of Volume fraction of the water. \'1\' represents water and \'0\' represents air.

Animation of the volume fraction of water. it is not that great because of the poor mesh quality.

Case-2:

Below plot represents the residuals. Convergence is checked so the solution has stopped way before it has reached the 200 timesteps.

It is the contour of Volume fraction of the water. \'1\' represents water and \'0\' represents air. we got a detailed contour because of the mesh refinement made. As the simulation ran for longer time than case-1 we can observe the heavier fluid is settling down.

Animation of volume fraction of water.

Case-3:

Below plot represents the residuals. Convergence is checked so the solution has stopped way before it has reached the 200 timesteps. this simulation has reached convergence faster than case-2 eventhough there more cells in the domain.

This is a high resolution countour of the volume fraction of water as the mesh size is less and the no of cells in the domain are more.

Animation of volume fraction of water of mesh size 0.2

3. Atwood Number

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

where

= density of heavier fluid

= density of lighter fluid

In this case Atwood number = (998.2-1.225)/(998.2+1.225) =996.975/999.425 = 0.997.

For A close to 0, RT instability flows take the form of symmetric \"fingers\" of fluid; for A close to 1, the much lighter fluid \"below\" the heavier fluid takes the form of larger bubble-like plumes.

According to the Atwood number when the ligther fluid is below the heavier fluid it has to form bubbles using this we can validate our results sincee we have observed bubbles formation in the domain which are formed by the ligther fluid. This tells is that the simulation results are correct.